In wireless communications, doubly-selective channels are subject to time-varying fading and multi-path delays. Time-varying fading has a specific Doppler power spectrum, causing time-selective channels and inter-carrier interference (ICI). Multi-path delays have a certain delay power spectrum, causing frequency-selective channels and inter-symbol interference (ISI). This is a particular problem in a wideband wireless network with mobile transceivers, e.g. in vehicular communications networks and cellular networks.
Adaptive equalization and pilot symbols are typically used to compensate for ISI and ICI. For example, pilot symbols or training sequences are inserted in packets transmitted to a receiver in order to estimate channel state information (CSI) of wireless links. The estimated CSI is used for equalization of ISI and ICI. However, the pilot symbols inherently reduce the spectral efficiency and data transmission rate, given restricted resources of radio frequency that is suited for mobile communications. Moreover, the pilot symbols are only effective for slow fading channels. In addition, channel estimation to obtain instantaneous CSI is considerably difficult in doubly-selective fading channels due to the presence of both ISI and ICI, even with pilot symbols.
A basis expansion model (BEM) has been used to approximate singly-selective fading channels, i.e., either time-varying or multi-path channels, to realize semi-blind transmissions for mobile wireless communications, see, e.g., U.S. Pat. No. 7,280,604 B2. The BEM is used in conjunction with an iterative semi-blind equalizer based on an expectation-and-maximization (EM) procedure for joint data detection and CSI estimation. With help of a forward error correction (FEC) code, the BEM can realize quasi non-coherent communications with a small number of pilot symbols without performance degradation. However, that method still relies on pilot symbols to obtain an initial CSI estimation and FEC soft-decision decoding feedbacks, which can make a receiver more complicated.
Differential space-time coding (DSTC) has been used to realize non-coherent communications and to provide diversity gain to be robust against channel fading, see, e.g., U.S. Pat. No. 7,567,623 B2, U.S. Pat. No. 7,508,808 B2, and U.S. Pat. No. 7,397,866 B2. The non-coherent communications do not need to use any pilot symbols, and higher spectral efficiency can be maintained because of no communications overhead. DSTC is realized by Grassmannian orthogonal matrix, see, e.g., U.S. Pat. No. 6,801,579 B1, U.S. Pat. No. 7,864,876 B2, and US 2002/0163892 A1. However, DSTC does not perform well in fast time-varying channels, where channel coherence time is very short to exploit the orthogonal feature of the Grassmannian matrix. To solve this problem, the BEM and DSTC can be simultaneously used with generalized likelihood-ratio test (GLRT) equalization, see, e.g., U.S. Pat. No. 8,059,747. With BEM, non-coherent communications can be realized even for fast time-varying fading channels and for high-speed data rate transmissions. This method and system can be extended to doubly-selective fading channels by introducing multi-dimensional BEM, in which two different kinds of basis functions over time and frequency domain are used to deal with doubly-selective fading channels.
However, the BEM itself is susceptive to some hardware imperfections, such as timing offset and carrier offset. For example, the timing and carrier offsets can be caused by hardware impairments, such as inaccuracy of a clock circuit including, e.g., a crystal oscillator. Such offsets are undesired for a number of applications, and can be compensated for by a specially designed synchronization process. For example, U.S. Pat. No. 7,961,697 B2 describes a method for synchronizing clocks between the receiver and transmitter using pre-defined training sequences in synchronization packets. However, transmission of the synchronization packets causes additional communications overhead, which is an undesirable or even unacceptable for a number of applications.
Accordingly, there is a need for a method for non-coherent transmission for doubly-selective channels accounting for timing offset and carrier offset of the transmission without relying on any pilots or training sequences for synchronization and channel estimation.